The universal variational expansion for the non-relativistic three-bod\ y systems is explicitly constructed. This universal expansion can be used to per\ form highly accurate numerical computations of the bound state spectra in variou\ s three-body systems, including Coulomb three-body systems with arbitrary partic\ le masses and electric charges. Our main interest is related to the adiabatic th\ ree-body systems which contain one bound electron and two heavy nuclei of hydrog\ en isotopes: the protium p, deuterium d and tritium t. We also consider the anal\ ogous (model) hydrogen ion ∞H2+ with the two infinitely heavy nuclei.